Note to students: some of the tables are slightly distorted due to the conversion process, though in most case they are readable. If a clearer version of the a table is needed, ask Randy Souviney or Cori Ewing for a hard (printed) copy.
Chapter Five
Computers and Calculators In Mathematics Education
I Was Just Thinking...
Take any three-digit number and write it down twice to make a six-digit number (i.e., 123123). Use a calculator to show that the six-digit number is divisible by seven without a remainder. Then show that the six-digit number is divisible by eleven without remainder. Finally, show that the six-digit number is divisible by thirteen without a remainder. Can you explain why this works?
NCTM Standards On Computers and Calculators
... some aspects of doing mathematics have changed in the last decade. The computerÕs ability to process large sets of information has made quantification and the logical analysis of information possible in such areas as business, economics, linguistics, biology, medicine, and sociology. Change has been particularly great in the social and life sciences. In fact, quantitative techniques have permeated almost all intellectual disciplines. However, the fundamental mathematical ideas needed in these areas are not necessarily those studied in the traditional algebra-geometry-precalculus-calculus sequence, a sequence designed with engineering and physical science applications in mind. Because mathematics is a foundation discipline for other disciplines and grows in direct proportion to its utility, we believe that the curriculum for all students must provide opportunities to develop an understanding of mathematical models, structures, and simulations applicable to many disciplines...Because technology is changing mathematics and its uses, we believe thatÐ
¥ appropriate calculators should be available to students at all times;
¥ a computer should be available in every classroom for demonstration purposes;
¥ every student should have access to a computer for individual and group work;
¥ students should learn to use the computer as a tool for processing information and performing calculations to investigate and solve problems.
(NCTM Standards, pp. 7-8)
××××
The major topics addressed in Chapter 5 include:
¥ Uses of calculators in elementary mathematics instruction.
¥ Criteria for selecting classroom calculators.
¥ Instructional applications of computers.
¥ Characteristics of good educational software.
¥ How calculators and computers can be used by students to solve problems.
ENIAC, the first commercial digital computer, was developed in 1945 at the University of Pennsylvania as a direct outcome of the second world war. During the next two decades, the applications of computers in teaching and learning were explored. Due to the expense of computers at that time, these studies had little effect on classroom practice.
The current revolution in education computing is a by product of the Òspace raceÓ that made available low cost, miniaturized electronic components. The first microprocessor, a computer on a chip, was developed in 1971. Since then, significant advances in microcircuits and information storage has produced inexpensive desk-top computers that rival the capabilities of the largest computer in the 1971.
Over one-million personal computers are currently being used in elementary schools throughout the United States and most provide sets of calculators for student use. In 1990, the ÒtypicalÓ elementary school had about 20 personal computers for instructional use (Becker, 1990). Although approximately two-thirds of the total computer time is devoted to basic skills instruction, students are increasingly using them as ÒtoolsÓ to analyze data and word process. Inexpensive, hand-held programmable calculators are used in secondary classrooms to construct graphs and evaluate equations. By the start of the next century, we will see computers influencing our society as profoundly as the cultural revolution caused the invention of the Gutenberg printing press 400 years ago.
Instructional Uses of Calculators
The National Council of Teacher of Mathematics (NCTM) Standards (1989) recommended that programs at all levels take full advantage of calculators and computers in mathematics instruction. Based on a long series of research studies, NCTM issued a detailed Position Statement on Calculators in the Mathematics Classroom (1987). The report recommends that all students have an appropriate calculator for mathematics learning and that calculators be integrated into classwork, homework, and assessment. Ready access to calculators in school would reduce the amount of time spent on computation drills. This would allow increased emphasis on problem solving, mathematical reasoning, and applications.
Table 5.1 lists five important instructional uses for calculators and research studies which support each (NCTM, 1987).
Table 5.1
Instructional Uses of Calculators
Instructional Use
Research Support
Concentrate on the problem-solving process rather than on the calculations associated with problems.
Bitter and Hatfield (1992) found that the integration of calculators into the middle-school curriculum improved problem solving performance, especially for girls. (Also see Bartalo, 1983; Comstock & Demana, 1987.)
Gain access to mathematics beyond the studentsÕ level of computational skills.
Allinger (1985) found that low-achieving students participated more effectively in mathematics class with access to a calculator. (Also see Suydam, 1982 & VanDevender & Rice, 1984.)
Explore, develop, and reinforce concepts including estimation, computation, approximation, and number properties.
Dossey (1990) found that students can use Òfast pencilsÓ to more effectively develop mathematics concepts and procedures. (Also see Hope & Sherrill, 1987.)
Experiment with mathematical ideas and discover patterns.
Usiskin (1978) argues that the calculator can be an used effectively in exploratory activities and group investigations. (Also see Bell, 1978.)
Perform tedious computations that arise when working with real data in problem-solving situations.
Dick (1988) found that regular use of a calculator helps students judge the reasonableness of a solution. (Also see Suydam, 1982.)
Several states have taken the position that calculators should be available for instruction and testing of all children in grades K-12 (California State Department of Education, 1992; Carter & Leinwand, 1987). Calculators have been employed to improve studentsÕ understanding of basic concepts, provide additional skill practice, and support real problem solving in the classroom.
Learning to calculate accurately is an important part of growing up in our complex society. Many events in everyday life involve working with various kinds of numbers. Counting, making change, comparison shopping, figuring sales tax, filing income taxes, doing carpentry, and working in the garden all require an ability to manipulate numbers sensibly. It is not only necessary to estimate approximate answers and calculate accurately when required, but it is also important to have a good understanding of the underlying concepts in order to know when to apply the appropriate procedure. Getting the right answer does little good if you solve the wrong problem!
When inexpensive calculators first became available in the mid-1970s, there was concern about their potential negative effects on the development of children's computation skills and mathematics learning. Since the early 1980s, several hundred studies of classroom calculator-use have uncovered virtually no negative effects on mathematics learning (Driscoll, 1981; Suydam, 1982; Hembree & Dessart, 1986). In fact, research results have shown the calculator to be a useful tool to support the teaching of mathematics concepts, procedures, and problem solving at all grade levels (Bitter & Hatfield, 1992; Wheatley & Shumway, 1979).
Twenty-first century children will still need to learn the basic facts and computation algorithms. The understanding of number operations and facility in computing small numbers is fundamental to the efficient application of arithmetic to real world problems. However, solving problems involving fractions, decimals, or very large and small values can be facilitated by using a calculator. Students can focus more attention on understanding problems and determining appropriate solution procedures if a calculator is available to handle the routine calculations. Students may also be more willing to risk tentative solutions if they are not confronted with several minutes of tedious arithmetic in order to test their conjectures (for a sample activity, see the section ÒSolving Problems with CalculatorsÓ).
From a Different Angle (Sidebar Dialogue)
ST: I read that students are allowed to use calculators on standardized exams. Do you think that is a good idea?
CT:Well, it seems to be. The NCTM Standards are recommending that math lessons focus on problem solving at all grade levels. This means that accurate, paper-and-pencil computation will be stressed less than in the past. Calculators are necessary if we want our kids to spend their time understanding problem situations and working towards solutions. On an exam, if you want to test problem solving ability, calculators level the computation playing field.
ST: But wonÕt using calculators too early make it more likely that students will not develop a good understanding of number operations Ð place value, when to use addition or division in a problem?
CT:Sure if they are introduced indiscriminately. We will need to increase our emphasis on estimation and mental arithmetic at the same time. We donÕt want children entering 2 ´ 10 into a calculator to find the answer. One of the most important roles for calculators occurs after number concepts have been developed using manipulative activities. Calculators can then be used whenever accurate answers are needed for exercises that can not be easily accomplished mentally.
Calculator Applications. Many calculator activities and strategy games have been designed to give children practice discovering patterns, thinking logically, and checking results (Immerzeel and Ockenga, 1977; Bitter & Mikesell, 1990). For example, calculators can be effectively employed to
¥ Develop the concept of place-value (e.g., enter the value 68341 into a calculator, what number must be subtracted in order to leave a zero in the thousands place while all the other digits remain the same?).
¥ Carry out tedious calculations or those which exceed studentsÕ current level of ability (e.g., a second grader might recognize the need to add the costs of ten items purchased at the store but need help with the computation.).
¥ Practice estimation skills (e.g., enter 142 Ö 16, estimate the solution, then press = to check the accuracy.).
¥ Give instant feedback for basic-fact drills (e.g., enter 3 + 4, think of the answer, and press = to check solution.) Note: Inexpensive preprogrammed calculators are available which embed this activity in a game format.
¥ Enhance insight into why algorithms work (e.g., to demonstrate that 24 ´ 39 = 936, first calculate the subproducts 4 ´ 39 and 20 ´ 39, then add the results 156 + 780 = 936).
¥ Introduce new concepts such a percent, negative numbers, and square root (e.g., entering the subtraction exercise 4 -7= displays the solution -3).
¥ Play number-sequence strategy games (e.g., for two players sharing a calculator, see who can get to 21 first by starting at 0 and alternately adding 1 or 2 to the displayed sum).
Selecting a Calculator. Calculators with a range of features are available for less than five dollars each. Just like a pencil, every elementary student should have a calculator available during mathematics lessons. This is especially true for Grade 3-6. If calculators are not available at school, students can be encouraged to bring a calculator from home for their personal use in school and to take home for homework (make sure you have a secure place to store the calculators when they are not in use).
When selecting a calculator for primary aged children, consider the following criteria:
¥ Good quality keys and a large, easy-to-read, eight- to ten-digit display with a backspace key.
¥ Solar-powered with a backup battery.
¥ Pressing any operation key ( +, -, ´, Ö) when carrying out a chain of calculations should cause the pending operation to be executed (e.g., entering 2 + 3 + should display the value 5.).
¥ The availability of a memory register (memory + or -) which makes two-step computations easier because the result of an intermediate step can be stored and recalled later (e.g., the exercise, (3 + 4) ´ (5 + 6) = ? can be solved by pressing keys in the following sequence: 3 + 4 = (answer 7) Memory + 5 + 6 = (answer 11) ´ Memory Recall (MR) = (answer 77)).
¥ The availability of a constant mode, which means that the calculator will repeat an operation if the key is pressed two or more times (e.g., to show that 4 ´ 3 can be calculated by adding 4 + 4 + 4, press 4 + + + to display the answer 12.).
Additional desirable functions for Grade 4-6 students include:
¥ A square root key that causes the number in the display to be replaced with its square root (e.g., entering 25 and then pressing the square root key gives the answer 5.).
¥ A change sign key (+/-) that allows the user to change the sign of any number displayed (e.g., entering the value 3 +/- will display -3.).
¥ A reciprocal f(1,n) key that causes the number in the display to be replaced with its reciprocal (e.g., entering the value 4 followed by the reciprocal key gives the answer 0.25 or f(1,4).).
¥ Capable of calculating using standard fractions without conversion to decimal form.
¥ Capable of calculating integer quotients and displaying whole number remainders.
¥ Capable of using parentheses to specify the order of operations.
Calculator for Grades K-3. Select a calculator for Grades K-3 that has large keys and easy to read display. Several companies produce calculators with a protective slip cover that are packaged for schools in sets of thirty. Texas Instruments supplies the Educatorª Basic Overhead Calculator that makes it easy to demonstrate the matching TI-108 student calculator for an entire class.
Calculator for Grades 4-6. The Texas Instruments Math Explorerª is an example of a calculator specifically developed for use by Grade 4-8 students. Unique features enable the calculator to be used more flexibly for mathematics instruction. It is capable of calculating with fractions without conversion to decimals. Simplifying fractions can be accomplished by entering a common divisor for the numerator and denominator, or it will automatically reduce the fraction to lowest terms and show the user the common factor used in completing the operation. Another useful feature is the integer divide key. Division using this feature gives the result as a whole number quotient and a remainder, rather than the typical decimal solution. This facilitates division of time and other measures when solving problems (e.g., 16 feet Ö 5 = 3 and 1 foot remaining, rather than 16 Ö 5 = 3.2 as would be shown on a typical calculator). The Math Explorerª also has a handy backspace key, a fixed decimal mode for working with money, a percent key, memory keys, four exponent functions to find square roots and powers of 10, parentheses to help with order of operations, and runs on solar cells (see Figure 5.1). The Educatorª Intermediate Overhead Calculator which matches the keyboard layout of the Math Explorerª is also available for whole-group demonstrations.
Figure 5.1
New art of Math Explorerª calculator.
Solving Problems with Calculators
Calculators are useful problem-solving tools in the classroom. In the following example, children can use a calculator to complete routine calculations, freeing them to focus attention on the underlying structure of the problem.
Suppose you wish to construct a box from a 1 ´ 1 meter piece of cardboard. Boxes are formed by cutting identical squares from each corner and folding each side into position (see 5.2).
Figure 5.2
Insert art of square with corners cut out.
As you vary the size of the four identical squares cut from each corner, different boxes can be constructed. Which box will have the greatest volume? First, letÕs explore some questions students may ask as they try to understand this problem.
1.Will the boxes constructed using the method described will have different volumes?
2.How many different boxes can be constructed?
3.Is there a limit to how large the volume can get?
4.What is the largest square that can be removed from each corner and still make a box? The smallest?
5.How do you compute the volume of a box?
6.How accurate should the answer be?
To answer these questions, students could draw sketches and compute the volumes of the resulting boxes using a calculator or they could actually construct boxes using paper or cardboard until they are satisfied that the boxes do represent a range of volumes and that it is impossible to make them all. One possibility is to simplify the problem by allowing only squares with edges that are whole centimeters (1 cm, 2 cm, 3 cm, and so on) or multiples of 10 centimeters (10 cm, 20 cm, 30 cm, and so on) to be cut from the four corners. The resulting volumes could be computed using a calculator and the results compiled as in Table 5.2.